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A099308
Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.
28
0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 105, 107, 109, 113, 114, 118, 121, 126, 127, 129, 130
OFFSET
1,3
COMMENTS
The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.
For all n, A003415(a(n)) is also a term of the sequence. A351255 gives the nonzero terms as ordered by their position in A276086. - Antti Karttunen, Feb 14 2022
REFERENCES
See A003415.
LINKS
Victor Ufnarovski and Bo Ã…hlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
FORMULA
For all n >= 0, A328309(a(n)) = n. - Antti Karttunen, Feb 14 2022
EXAMPLE
18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 0.
MATHEMATICA
dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; nLim=200; lst={1}; i=1; While[i<=Length[lst], currN=lst[[i]]; pre=Intersection[Flatten[Position[d1, currN]], Range[nLim]]; pre=Complement[pre, lst]; lst=Join[lst, pre]; i++ ]; Union[lst]
PROG
(PARI)
\\ The following program would get stuck in nontrivial loops. However, we assume that the conjecture 3 in Ufnarovski & Ã…hlander paper holds ("The differential equation n^(k) = n has only trivial solutions p^p for primes p").
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
isA099308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n)); \\ Antti Karttunen, Feb 14 2022
CROSSREFS
Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099309 (complement, numbers whose k-th arithmetic derivative is nonzero for all k), A351078 (first noncomposite reached when iterating the derivative from these numbers), A351079 (the largest term on such paths).
Cf. A328308, A328309 (characteristic function and their partial sums), A341999 (1 - charfun).
Cf. A276086, A328116, A351255 (permutation of nonzero terms), A351257, A351259, A351261, A351072 (number of prime(k)-smooth terms > 1).
Cf. also A256750 (number of iterations needed to reach either 0 or a number with a factor of the form p^p), A327969, A351088.
Union of A359544 and A359545.
Sequence in context: A248565 A065896 A358215 * A359544 A074235 A325366
KEYWORD
nonn
AUTHOR
T. D. Noe, Oct 12 2004
STATUS
approved